STA 402S 1998: SAS Lesson 4
Castle Bakery Example from Chapter 19
The file castle.dat looks like this.
1 1 47 1 1 43 2 1 62 2 1 68 3 1 41 3 1 39 1 2 46 1 2 40 2 2 67 2 2 71 3 2 42 3 2 46
Here is the command file.
/************* castle1.sas *********************
Two-way ANOVA SAS example with dummy var coding:
See Section 19.4; data on P. 818
***************************************************/
options linesize = 79;
proc format; /* value labels used in data step below */
value htfmt 1 = 'Bottom' 2 = 'Middle' 3 = 'Top';
value widfmt 1 = 'Regular' 2 = 'Wide';
data bake;
infile 'castle.dat';
input height width sales;
label height = 'Display Height'
width = 'Display Width'
sales = 'Sales in cases';
format height htfmt.;
format width widfmt.; /* Asssociate variables with print formats */
/* Effect dummy variable coding */
if width = 1 then w1 = 1; else w1 = -1;
if height = 1 then h1 = 1;
else if height = 3 then h1 = -1;
else h1 = 0;
if height = 2 then h2 = 1;
else if height = 3 then h2 = -1;
else h2 = 0;
h1w1 = h1*w1 ; h2w1 = h2*w1; /* Interactions */
/* Cell means coding: indicators are named after their parameters. */
if height = 1 and width = 1 then mu11 = 1 ; else mu11 = 0;
if height = 1 and width = 2 then mu12 = 1 ; else mu12 = 0;
if height = 2 and width = 1 then mu21 = 1 ; else mu21 = 0;
if height = 2 and width = 2 then mu22 = 1 ; else mu22 = 0;
if height = 3 and width = 1 then mu31 = 1 ; else mu31 = 0;
if height = 3 and width = 2 then mu32 = 1 ; else mu32 = 0;
/* Make combination variable */
hwcombo = 10*height + width;
proc freq;
tables height * width / nopercent norow nocol;
tables hwcombo * height width / nopercent norow nocol;
proc glm; /* glm does the dummy vars for us */
class height width;
model sales = height width height*width;
means height width height*width;
means height / alpha=.05 tukey bon scheffe cldiff;
/* Why didn't I bother to do it for shelf width? */
proc reg; /* Now do it with dummy var regression using effect coding */
model sales = h1 h2 w1 h1w1 h2w1;
height: test h1=h2=0;
width: test w1 = 0;
h_by_w: test h1w1 = h2w1 = 0;
proc reg; /* Now with cell means coding */
model sales = mu11 -- mu32 / noint ;
height: test mu11+mu12=mu21+mu22=mu31+mu32;
width: test mu11+mu21+mu31=mu12+mu22+mu32;
h_by_w: test mu11-mu12 = mu21-mu22 = mu31-mu32;
proc glm; /* Combination variables are good for comparing cell means */
class hwcombo;
model sales=hwcombo;
means hwcombo / tukey; /* Could have used cldiff option */
Part of the lst file:
TABLE OF HEIGHT BY WIDTH
HEIGHT(Display Height)
WIDTH(Display Width)
Frequency|Regular |Wide | Total
---------+--------+--------+
Bottom | 2 | 2 | 4
---------+--------+--------+
Middle | 2 | 2 | 4
---------+--------+--------+
Top | 2 | 2 | 4
---------+--------+--------+
Total 6 6 12
TABLE OF HWCOMBO BY HEIGHT
HWCOMBO HEIGHT(Display Height)
Frequency|Bottom |Middle |Top | Total
---------+--------+--------+--------+
11 | 2 | 0 | 0 | 2
---------+--------+--------+--------+
12 | 2 | 0 | 0 | 2
---------+--------+--------+--------+
21 | 0 | 2 | 0 | 2
---------+--------+--------+--------+
22 | 0 | 2 | 0 | 2
---------+--------+--------+--------+
31 | 0 | 0 | 2 | 2
---------+--------+--------+--------+
32 | 0 | 0 | 2 | 2
---------+--------+--------+--------+
Total 4 4 4 12
General Linear Models Procedure
Class Level Information
Class Levels Values
HEIGHT 3 Bottom Middle Top
WIDTH 2 Regular Wide
Number of observations in data set = 12
General Linear Models Procedure
Dependent Variable: SALES Sales in cases
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 5 1580.0000000 316.0000000 30.58 0.0003
Error 6 62.0000000 10.3333333
Corrected Total 11 1642.0000000
R-Square C.V. Root MSE SALES Mean
0.962241 6.303040 3.2145503 51.000000
Source DF Type I SS Mean Square F Value Pr > F
HEIGHT 2 1544.0000000 772.0000000 74.71 0.0001
WIDTH 1 12.0000000 12.0000000 1.16 0.3226
HEIGHT*WIDTH 2 24.0000000 12.0000000 1.16 0.3747
Source DF Type III SS Mean Square F Value Pr > F
HEIGHT 2 1544.0000000 772.0000000 74.71 0.0001
WIDTH 1 12.0000000 12.0000000 1.16 0.3226
HEIGHT*WIDTH 2 24.0000000 12.0000000 1.16 0.3747
General Linear Models Procedure
Level of ------------SALES------------
HEIGHT N Mean SD
Bottom 4 44.0000000 3.16227766
Middle 4 67.0000000 3.74165739
Top 4 42.0000000 2.94392029
Level of ------------SALES------------
WIDTH N Mean SD
Regular 6 50.0000000 12.0664825
Wide 6 52.0000000 13.4313067
Level of Level of ------------SALES------------
HEIGHT WIDTH N Mean SD
Bottom Regular 2 45.0000000 2.82842712
Bottom Wide 2 43.0000000 4.24264069
Middle Regular 2 65.0000000 4.24264069
Middle Wide 2 69.0000000 2.82842712
Top Regular 2 40.0000000 1.41421356
Top Wide 2 44.0000000 2.82842712
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: SALES
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 6 MSE= 10.33333
Critical Value of Studentized Range= 4.339
Minimum Significant Difference= 6.9743
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
HEIGHT Confidence Between Confidence
Comparison Limit Means Limit
Middle - Bottom 16.026 23.000 29.974 ***
Middle - Top 18.026 25.000 31.974 ***
Bottom - Middle -29.974 -23.000 -16.026 ***
Bottom - Top -4.974 2.000 8.974
Top - Middle -31.974 -25.000 -18.026 ***
Top - Bottom -8.974 -2.000 4.974
Compare Tukey confidence intervals on P. 858.
Now regression: First effect coding
Model: MODEL1
Dependent Variable: SALES Sales in cases
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 5 1580.00000 316.00000 30.581 0.0003
Error 6 62.00000 10.33333
C Total 11 1642.00000
Root MSE 3.21455 R-square 0.9622
Dep Mean 51.00000 Adj R-sq 0.9308
C.V. 6.30304
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 51.000000 0.92796073 54.959 0.0001
H1 1 -7.000000 1.31233465 -5.334 0.0018
H2 1 16.000000 1.31233465 12.192 0.0001
W1 1 -1.000000 0.92796073 -1.078 0.3226
H1W1 1 2.000000 1.31233465 1.524 0.1783
H2W1 1 -1.000000 1.31233465 -0.762 0.4749
Dependent Variable: SALES
Test: HEIGHT Numerator: 772.0000 DF: 2 F value: 74.7097
Denominator: 10.33333 DF: 6 Prob>F: 0.0001
Dependent Variable: SALES
Test: WIDTH Numerator: 12.0000 DF: 1 F value: 1.1613
Denominator: 10.33333 DF: 6 Prob>F: 0.3226
Dependent Variable: SALES
Test: H_BY_W Numerator: 12.0000 DF: 2 F value: 1.1613
Denominator: 10.33333 DF: 6 Prob>F: 0.3747
Compare, from GLM (earlier)
Source DF Type I SS Mean Square F Value Pr > F HEIGHT 2 1544.0000000 772.0000000 74.71 0.0001 WIDTH 1 12.0000000 12.0000000 1.16 0.3226 HEIGHT*WIDTH 2 24.0000000 12.0000000 1.16 0.3747
And now cell means coding
Model: MODEL1
NOTE: No intercept in model. R-square is redefined.
Dependent Variable: SALES Sales in cases
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 6 32792.00000 5465.33333 528.903 0.0001
Error 6 62.00000 10.33333
U Total 12 32854.00000
Root MSE 3.21455 R-square 0.9981
Dep Mean 51.00000 Adj R-sq 0.9962
C.V. 6.30304
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
MU11 1 45.000000 2.27303028 19.797 0.0001
MU12 1 43.000000 2.27303028 18.917 0.0001
MU21 1 65.000000 2.27303028 28.596 0.0001
MU22 1 69.000000 2.27303028 30.356 0.0001
MU31 1 40.000000 2.27303028 17.598 0.0001
MU32 1 44.000000 2.27303028 19.357 0.0001
Note the overall ANOVA is different: The null hypothesis here is that all means equal zero, not just that they are equal. Also, comparing parameter estimates to sample cell means from earlier, ...
Level of Level of ------------SALES------------
HEIGHT WIDTH N Mean SD
Bottom Regular 2 45.0000000 2.82842712
Bottom Wide 2 43.0000000 4.24264069
Middle Regular 2 65.0000000 4.24264069
Middle Wide 2 69.0000000 2.82842712
Top Regular 2 40.0000000 1.41421356
Top Wide 2 44.0000000 2.82842712
But the tests for main effects and interactions are the same.
Dependent Variable: SALES
Test: HEIGHT Numerator: 772.0000 DF: 2 F value: 74.7097
Denominator: 10.33333 DF: 6 Prob>F: 0.0001
Dependent Variable: SALES
Test: WIDTH Numerator: 12.0000 DF: 1 F value: 1.1613
Denominator: 10.33333 DF: 6 Prob>F: 0.3226
Dependent Variable: SALES
Test: H_BY_W Numerator: 12.0000 DF: 2 F value: 1.1613
Denominator: 10.33333 DF: 6 Prob>F: 0.3747
Now the combination variable. Really, this is most useful when the interaction is significant.
General Linear Models Procedure
Class Level Information
Class Levels Values
HWCOMBO 6 11 12 21 22 31 32
Number of observations in data set = 12
Dependent Variable: SALES Sales in cases
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 5 1580.0000000 316.0000000 30.58 0.0003
Error 6 62.0000000 10.3333333
Corrected Total 11 1642.0000000
R-Square C.V. Root MSE SALES Mean
0.962241 6.303040 3.2145503 51.000000
Source DF Type I SS Mean Square F Value Pr > F
HWCOMBO 5 1580.0000000 316.0000000 30.58 0.0003
Source DF Type III SS Mean Square F Value Pr > F
HWCOMBO 5 1580.0000000 316.0000000 30.58 0.0003
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: SALES
NOTE: This test controls the type I experimentwise error rate, but
generally has a higher type II error rate than REGWQ.
Alpha= 0.05 df= 6 MSE= 10.33333
Critical Value of Studentized Range= 5.628
Minimum Significant Difference= 12.793
Means with the same letter are not significantly different.
Tukey Grouping Mean N HWCOMBO
A 69.000 2 22
A
A 65.000 2 21
B 45.000 2 11
B
B 44.000 2 32
B
B 43.000 2 12
B
B 40.000 2 31